Question:`A = (x^2 + 3x - 4)/(x^2 + 7x + 12), B = (x^2 + 2x - 3)/(x^2 + 6x - 7),` `C = (x^2 + 12x + 35)/(x^2 + 4x - 5)` তিনটি বীজগাণিতিক রাশি। ক. A কে লঘিষ্ট আকারে প্রকাশ করো। খ. A + B কে সরল করো। গ. দেখাও যে, `B xx C -: (x^2 - 9)/(x - 1) = 1/(x - 3)`
Answer ক.`A = (x^2 + 3x - 4)/(x^2 + 7x + 12)` `= (x^2 + 4x - x - 4)/(x^2 + 4x + 3x + 12)` `= (x(x + 4) - 1 (x + 4))/(x(x + 4) + 3(x + 4))` `= ((x + 4)(x - 1))/((x + 4)(x + 3))` `= (x - 1)/(x + 3)` (Ans) খ. `A + B = (x^2 + 3x - 4)/(x^2 + 7x + 12) + (x^2 + 2x - 3)/(x^2 + 6x - 7)` `= (x - 1)/(x + 3) + (x^2 + 3x - x - 3)/(x^2 + 7x - x - 7)` [ক থেকে] `= (x - 1)/(x + 3) + (x(x + 3) - 1(x + 3))/(x(x + 7) - 1(x + 7))` `= (x - 1)/(x + 3) + ((x + 3) (x - 1))/((x + 7)(x - 1))` `= (x - 1)/(x + 3) + (x + 3)/(x + 7)` `= ((x - 1)(x + 7) + (x + 3)^2)/((x + 3) (x + 7))` `= (x^2 + 7x - x - 7 + x^2 + 6x + 9)/(x^2 + (3 + 7)x + 3.7)` `= (x^2 + 6x - 7 + x^2 + 6x + 9)/(x^2 + 10x + 21)` `= (2x^2 + 12x + 2)/(x^2 + 10x + 21)` (Ans) গ. বামপক্ষ` = B xx C -: (x^2 - 9)/(x - 1)` `= (x^2 + 2x - 3)/(x^2 + 6x - 7) xx (x^2 + 12x + 35)/(x^2 + 4x - 5) -: (x^2 - 9)/(x - 1)` `= (x^2 + 2x - 2)/(x^2 + 7x - x - 7) xx (x^2 + 7x + 5x + 35)/(x^2 + 5x - x - 5) -: (x - 1)/(x^2 - 9)` `= (x^2 + 2x - 2)/(x(x + 7) - 1(x - 7)) xx (x(x + 7) + 5(x + 7))/(x(x + 5) - 1(x + 5)) xx (x - 1)/(x^2 - 3^2)` `= (x^2 + 2x - 2)/((x + 7) (x - 1)) xx ((x + 7)(x + 5))/((x + 5)(x - 1)) xx (x - 1)/((x + 3)(x - 3))` `= (x^2 + 2x - 3)/((x - 1)(x + 3)(x - 3))` `= (x^2 + 2x - 3)/((x^2 + 3x - x - 3)(x - 3))` `= (x^2 + 2x - 3)/((x^2 + 2x - 3)(x - 3))` `= 1/(x - 3)` = ডানপক্ষ `:. B xx C -: (x^2 - 9)/(x - 1) ` `= 1/(x - 3)` (প্রমাণিত)
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`A = (x^2 + 3x - 4)/(x^2 + 7x + 12), B = (x^2 + 2x - 3)/(x^2 + 6x - 7),` `C = (x^2 + 12x + 35)/(x^2 + 4x - 5)` tinti bijoganitik rashi. ka. A ke lghisht akare prokasho karo. kh. A + B ke shorol karo. ga. dekhao je, `B xx C -: (x^2 - 9)/(x - 1) = 1/(x - 3)`